**1. Introduction**

A well-founded risk assessment is a mandatory stage in the effective implementation of almost any project. This chapter discusses credit risk assessments. Credit risk assessment includes determining whether or not to give a loan to a borrower and what is the probability of bankruptcy or inability to service a loan due to financial problems. Despite the fact that during this process the lender gets a lot of information from the borrower, there is no unambiguous rule for decision-making.

There are various models for assessing these risks. Before the 2008 financial crisis, emphasis was placed on developing models for assessing the financial stability of borrowers. But when, after this crisis, many companies faced a significant risk of bankruptcy, the vector of model developers took the direction to developing effective forecast models.

Existing forecasting models can be classified into two main groups: statistical and theoretical. A description of these models can be found in the literature (see, e.g., [1]). However, in a number of cases, these models are unacceptable. Basically, they are not suitable for assessing credit risks for corporations in developing countries, as well as for assessing the risks of lending to investment projects.

Let us dwell very briefly on the reasons for the unsuitability of these models for corporations in developing countries (a more detailed analysis can be found, e.g., in [2]). For statistical models:

• An insufficient defaulting history does not provide a relevant background for assessment of the credit risk.

Lotfi A. Zadeh was the first to propose a generalization of the range of values of the membership function {0, 1} to the closed interval [0; 1]. Thus, the value of the membership function can be any real number, starting from zero and ending with unity [3]. Such sets were called "fuzzy" sets. By gradually developing the proposed approach, Zadeh introduced the concept of a fuzzy linguistic variable, which was able to model mathematically linguistic variables [4]. For example, Zadeh made it possible to express mathematically the following linguistic notions: "childhood, young, middle-aged, old." Zadeh also introduced the concept of fuzzy relations and

*A New Approach for Assessing Credit Risks under Uncertainty*

*DOI: http://dx.doi.org/10.5772/intechopen.93285*

As noted above, in our study we cannot do without expert evaluations. Since subjectivity, vagueness, and imprecision influence the assessments of experts, the use of fuzzy set theory seems to be an effective tool for our research work (see, e.g., [5, 6]). In [2], to assess credit risks, we used precisely fuzzy relations. In the present work, we propose to use trapezoidal fuzzy numbers as a form of presentation of experts' estimates. The rationale for this choice will be given in the third section. The chapter consists of six sections. The second section includes all the necessary

information to understand the material. The theoretical basis of the offered

cal application of the introduced approach. The sixth and the final section

**2. Essential notions and theoretical background**

approach is laid out, and some theoretical results are given. In the third section, the approach to assessment of credit risk is introduced and discussed. In particular, the rationale for the choice of fuzzy trapezoidal numbers as a form for the presentation of experts' estimates is given. The fourth section looks at the algorithm for realization of the proposed approach. The fifth section contains toy example of the practi-

In conclusion, we note that when reading a chapter, the reader is not required to have knowledge of higher mathematics, but only elementary knowledge of arithmetic, algebra, and geometry. Nevertheless, we will try to explain meaningfully mathematical symbols and concepts that may be unfamiliar to the reader.

In the introduction, we gave a substantive description of a fuzzy set; now we

**Definition 1.** An ordered pair f g *x*, *μ*ð Þ *x* , where*x*∈ *X*, *μ* : *X* ! ½ � 0, 1 is called a

Here *X* is the universal set of real numbers (universe), *μ*ð Þ *x* is the membership function of fuzzy set, and *μ* : *X* ! ½ � 0, 1 means that the membership function takes

An important special case of fuzzy sets is fuzzy numbers. A *fuzzy number* is a fuzzy subset of the universal set of real numbers that has a normal and convex membership function, that is, such that: (a) there is an element of the universe in which the membership function is equal to one, and also (b) when deviating from

In this chapter we will deal with trapezoidal fuzzy numbers. Almost all the results given in this and third sections are, with minor modifications, taken from

We denote trapezoidal fuzzy numbers in *<sup>X</sup>* by *<sup>R</sup>*<sup>~</sup> <sup>¼</sup> ð Þ *<sup>a</sup>*, *<sup>b</sup>*,*c*, *<sup>d</sup>* , 0<sup>&</sup>lt; *<sup>a</sup>*<sup>≤</sup> *<sup>b</sup>*<sup>≤</sup> *<sup>c</sup>*≤*d*. The membership function's graph is a trapezoid with vertices (*a*; 0), (*b*; 1), (*c*; 1), and (*d*; 0). We denote by <sup>Ψ</sup>ð Þ¼ *<sup>X</sup> <sup>R</sup>*~*<sup>i</sup>* <sup>¼</sup> *ai*, *bi*,*ci* ð Þ , *di* , *<sup>i</sup>* <sup>∈</sup> the set of all trapezoidal fuzzy

The determinations of some operations on trapezoidal fuzzy numbers are given

its maximum left or right, membership function does not increase.

[7]; therefore we will refrain from further citation.

basic operations on them.

summarize the chapter.

give its mathematical description.

values from the interval [0; 1] for all *x*.

numbers in the universe *X*.

fuzzy set.

below.

**183**

• Problems arise with classification process too: a borrower may have so-called partial default status.

For theoretical models:

• A potential problem that could arise when applying this model is that majority of companies'stock is not traded on a stock exchange. In this case, a rapid evaluation of the market value of the assets is difficult.

It is very important to note that companies whose shares are not traded on the stock exchange are not so rare in developed countries.

Let us consider in more detail the credit risks when financing investment projects. It is known that an investment project involves planning over time of three main cash flows: investment, current (operating) expenses, and income.

When implementing an investment project, the investor never has a comprehensive risk assessment, since frequent changes in the dynamically developing world cannot be foreseen. Therefore, there is an unforeseen circumstance not taken into account by the project (e.g., a catastrophe), which nevertheless happened and disrupt the investment process. At the same time, the investor must be as informed as possible in order to assess the risk of his investment decisions both at the stage of project development and during the investment process itself. In addition, it is important to keep in mind that prices and volumes of products sold, as well as cash values for materials, raw materials, and other goods and services, in the future can radically differ from their expected values at the time of planning the investment project.

Thus, the incompleteness and uncertainty of the information significantly affects the effectiveness of the investment project and often poses insurmountable risks. Therefore, a project considered to be profitable, in fact, may be losing. This may occur due to the risk of deviation of the values of the design parameters from the actual values or due to the complete neglect of any factors.

Based on the foregoing and world practice, models for assessing any object of study, including risks, are suitable only if there is a sufficiently large statistical base (general population). Consequently, these methods do not result in cases under uncertainty. What to do in the absence of such statistics? The accumulated experience shows that the only way out in this case is to use expert estimates. Thus, we come to the process of group decision-making. Decision-making processes are used in quite a variety of applications. The inherent property of these processes is to represent the transformation of individual opinions of experts to the resulting one.

First you need to perform the parameterization process, i.e., identify parameters that experts should evaluate. Denote the set of selected parameters by *P* ¼ *pi* , *<sup>i</sup>* <sup>¼</sup> 1, *<sup>n</sup>*. Next, you need to determine in what form and on what scale experts will evaluate the values of the selected parameters. Then generate risk criteria and aggregate expert assessments to make a decision according to the assigned criteria. A very important point is the determination of the form for expert evaluations. Here we propose to consider expert estimates in the form of fuzzy sets.

In 1965, a professor at California University (Berkeley) Lotfi A. Zadeh published a paper "Fuzzy Sets" that gave a birth to the modeling of human intellectual activity and allowed for new interpretations of some mathematical theories. According to classical mathematics, an object either belongs to some set or not, so the characteristic function of an ordinary set is defined as {0, 1} (if an element does not belong to the set, then 0, and if it does, then 1).

*A New Approach for Assessing Credit Risks under Uncertainty DOI: http://dx.doi.org/10.5772/intechopen.93285*

• An insufficient defaulting history does not provide a relevant background for

• Problems arise with classification process too: a borrower may have so-called

• A potential problem that could arise when applying this model is that majority of companies'stock is not traded on a stock exchange. In this case, a rapid

It is very important to note that companies whose shares are not traded on the

Let us consider in more detail the credit risks when financing investment projects. It is known that an investment project involves planning over time of three

When implementing an investment project, the investor never has a comprehensive risk assessment, since frequent changes in the dynamically developing world cannot be foreseen. Therefore, there is an unforeseen circumstance not taken into account by the project (e.g., a catastrophe), which nevertheless happened and disrupt the investment process. At the same time, the investor must be as informed as possible in order to assess the risk of his investment decisions both at the stage of project development and during the investment process itself. In addition, it is important to keep in mind that prices and volumes of products sold, as well as cash values for materials, raw materials, and other goods and services, in the future can radically differ from their expected values at the time of planning the investment project. Thus, the incompleteness and uncertainty of the information significantly affects the effectiveness of the investment project and often poses insurmountable risks. Therefore, a project considered to be profitable, in fact, may be losing. This may occur due to the risk of deviation of the values of the design parameters from

Based on the foregoing and world practice, models for assessing any object of study, including risks, are suitable only if there is a sufficiently large statistical base (general population). Consequently, these methods do not result in cases under uncertainty. What to do in the absence of such statistics? The accumulated experience shows that the only way out in this case is to use expert estimates. Thus, we come to the process of group decision-making. Decision-making processes are used in quite a variety of applications. The inherent property of these processes is to represent the transformation of individual opinions of experts to the resulting one. First you need to perform the parameterization process, i.e., identify parameters

that experts should evaluate. Denote the set of selected parameters by *P* ¼

Here we propose to consider expert estimates in the form of fuzzy sets.

the set, then 0, and if it does, then 1).

 , *<sup>i</sup>* <sup>¼</sup> 1, *<sup>n</sup>*. Next, you need to determine in what form and on what scale experts will evaluate the values of the selected parameters. Then generate risk criteria and aggregate expert assessments to make a decision according to the assigned criteria. A very important point is the determination of the form for expert evaluations.

In 1965, a professor at California University (Berkeley) Lotfi A. Zadeh published a paper "Fuzzy Sets" that gave a birth to the modeling of human intellectual activity and allowed for new interpretations of some mathematical theories. According to classical mathematics, an object either belongs to some set or not, so the characteristic function of an ordinary set is defined as {0, 1} (if an element does not belong to

main cash flows: investment, current (operating) expenses, and income.

evaluation of the market value of the assets is difficult.

the actual values or due to the complete neglect of any factors.

stock exchange are not so rare in developed countries.

assessment of the credit risk.

partial default status.

*Banking and Finance*

For theoretical models:

*pi*

**182**

Lotfi A. Zadeh was the first to propose a generalization of the range of values of the membership function {0, 1} to the closed interval [0; 1]. Thus, the value of the membership function can be any real number, starting from zero and ending with unity [3]. Such sets were called "fuzzy" sets. By gradually developing the proposed approach, Zadeh introduced the concept of a fuzzy linguistic variable, which was able to model mathematically linguistic variables [4]. For example, Zadeh made it possible to express mathematically the following linguistic notions: "childhood, young, middle-aged, old." Zadeh also introduced the concept of fuzzy relations and basic operations on them.

As noted above, in our study we cannot do without expert evaluations. Since subjectivity, vagueness, and imprecision influence the assessments of experts, the use of fuzzy set theory seems to be an effective tool for our research work (see, e.g., [5, 6]).

In [2], to assess credit risks, we used precisely fuzzy relations. In the present work, we propose to use trapezoidal fuzzy numbers as a form of presentation of experts' estimates. The rationale for this choice will be given in the third section.

The chapter consists of six sections. The second section includes all the necessary information to understand the material. The theoretical basis of the offered approach is laid out, and some theoretical results are given. In the third section, the approach to assessment of credit risk is introduced and discussed. In particular, the rationale for the choice of fuzzy trapezoidal numbers as a form for the presentation of experts' estimates is given. The fourth section looks at the algorithm for realization of the proposed approach. The fifth section contains toy example of the practical application of the introduced approach. The sixth and the final section summarize the chapter.

In conclusion, we note that when reading a chapter, the reader is not required to have knowledge of higher mathematics, but only elementary knowledge of arithmetic, algebra, and geometry. Nevertheless, we will try to explain meaningfully mathematical symbols and concepts that may be unfamiliar to the reader.
